Definitions

# Controlled NOT gate

The Controlled NOT gate (also C-NOT or CNOT) is a quantum gate that is an essential component in the construction of a quantum computer. It can be used to disentangle EPR states. Specifically, any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations.

## Operation

The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is 1.

Before After
Control Target Control Target
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0

The resulting value of the second qubit corresponds to the result of a classical XOR gate.

The CNOT gate can be represented by the matrix:

$operatorname\left\{CNOT\right\} = begin\left\{bmatrix\right\} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 0 & 1 0 & 0 & 1 & 0 end\left\{bmatrix\right\}.$

The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state and into the vibrational state of the ion within the trap. At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.

In addition to a regular Controlled NOT gate, one could construct a Function-Controlled NOT gate, which accepts an arbitrary number n+1 of qubits as input, where n+1 is greater than or equal to 2 (a quantum register). This gate flips the last qubit of the register if and only if a built-in function, with the first n qubits as input, returns a 1. The Function-Controlled NOT gate is an essential element of the Deutsch-Jozsa algorithm.

## Proof of operation

Let $left\left\{ |0rangle =! begin\left\{bmatrix\right\} 1 0 end\left\{bmatrix\right\}, |1rangle =! begin\left\{bmatrix\right\} 0 1 end\left\{bmatrix\right\} right\right\}$ be the orthonormal basis (using Bra-ket notation).

Let $| psi rangle = a | 0 rangle + b | 1 rangle =! begin\left\{bmatrix\right\} a b end\left\{bmatrix\right\}$.

Let $| phi rangle = b | 0 rangle + a | 1 rangle =! begin\left\{bmatrix\right\} b a end\left\{bmatrix\right\}$ be the flip qubit of $| psi rangle$.

Recall that $|alpharangle!otimes!|betarangle$

`                 = |alpharangle |betarangle`
`                 = |alpha,betarangle .`

### When control qubit is 0

First, we shall prove that $operatorname\left\{CNOT\right\} |0,psirangle = |0,psirangle$:

It's not difficult to verify that $|0,psirangle = begin\left\{bmatrix\right\} a b 0 0 end\left\{bmatrix\right\}$.

Then $operatorname\left\{CNOT\right\} |0,psirangle = begin\left\{bmatrix\right\} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 0 & 1 0 & 0 & 1 & 0 end\left\{bmatrix\right\} begin\left\{bmatrix\right\} a b 0 0 end\left\{bmatrix\right\} = a begin\left\{bmatrix\right\} 1 0 0 0 end\left\{bmatrix\right\} + b begin\left\{bmatrix\right\} 0 1 0 0 end\left\{bmatrix\right\}$.

As we can see $|0,0rangle = begin\left\{bmatrix\right\} 1 0 0 0 end\left\{bmatrix\right\}$ and $|0,1rangle = begin\left\{bmatrix\right\} 0 1 0 0 end\left\{bmatrix\right\}$, using these on the equation above gives

begin\left\{align\right\} operatorname\left\{CNOT\right\} |0,psirangle

`   & = a |0,0rangle + b |0,1rangle `
`   & = |0rangle left(a |0rangle + b |1rangle right) `
`   & = |0,psirangle `
end{align}

Therefore CNOT doesn't flip the $|psirangle$ qubit if the first qubit is 0.

### When control qubit is 1

Now, we shall prove that $operatorname\left\{CNOT\right\} |1,psirangle = |1,phirangle$, which means that the CNOT gate flips the $|psirangle$ qubit.

Similarly to the first demonstration, we have $|1,psirangle = begin\left\{bmatrix\right\} 0 0 a b end\left\{bmatrix\right\}$.

Then $operatorname\left\{CNOT\right\} |1,psirangle = begin\left\{bmatrix\right\} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 0 & 1 0 & 0 & 1 & 0 end\left\{bmatrix\right\} begin\left\{bmatrix\right\} 0 0 a b end\left\{bmatrix\right\} = a begin\left\{bmatrix\right\} 0 0 0 1 end\left\{bmatrix\right\} + b begin\left\{bmatrix\right\} 0 0 1 0 end\left\{bmatrix\right\}$

As we can see that $|1,1rangle = begin\left\{bmatrix\right\} 0 0 0 1 end\left\{bmatrix\right\}$ and $|1,0rangle = begin\left\{bmatrix\right\} 0 0 1 0 end\left\{bmatrix\right\}$, using these on the equation above gives

begin\left\{align\right\} operatorname\left\{CNOT\right\} |1,psirangle

`    & = a |1,1rangle + b |1,0rangle `
`    & = |1rangle left(a |1rangle  + b |0rangle right) `
`    & = |1,phirangle `
end{align}

Therefore the CNOT gate flips the $|psirangle$ qubit into $|phirangle$ if the control qubit is set to 1. A simple way to observe this is to multiply the CNOT matrix by a column vector, noticing that the operation on the first bit is identity, and a NOT gate on the second bit.

## References

• Nielsen, Michael A. & Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 0-521-63235-8.
• Monroe, C. & Meekhof, D. & King, B. & Itano, W. & Wineland, D. (1995). "Demonstration of a Fundamental Quantum Logic Gate". Physical Review Letters 75 (75): 4714–4717.

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